Question

If a line segment $A M=a$ moves in the plane $X O Y$ remaining parallel to $O X$ so that the left end point $A$ slides along the circle $x^{2}+y^{2}=a^{2}$, the locus of $M$ is

• A. $x^{2}+y^{2}=4 a^{2}$
• B. $x^{2}+y^{2}=2 a x$
• C. $x^{2}+y^{2}=2 a y$
• D. $x^{2}+y^{2}-2 a x-2 a y=0$

Answer 1

Answer: (B)

Let the coordinates of $A$ be $(x, y)$ and $M$ be $(\alpha, \beta)$
Since $A M$ is parallel to $O X$,

$\alpha=x+a$ and $\beta=y$
$ \Rightarrow x=\alpha-a$ and $y=\beta$
As $A(x, y)$ lies on the circle $x^{2}+y^{2}=a^{2}$, we have
$(\alpha-a)^{2}+\beta^{2}=a^{2} $
$\Rightarrow \alpha^{2}-2 a \alpha+\beta^{2}=0$
$\Rightarrow$ locus of $M(\alpha, \beta)$ is
$x^{2}+y^{2}=2 a x$.